The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors

Abstract

In this paper we discuss the notion of singular vector tuples of a complex-valued $$d$$ d -mode tensor of dimension $$m_1\times \cdots \times m_d$$ m 1 × ? × m d . We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous results for the homogeneous pencil eigenvalue problem for cubic tensors, i.e., $$m_1=\cdots =m_d$$ m 1 = ? = m d . We show the uniqueness of best approximations for almost all real tensors in the following cases: rank-one approximation; rank-one approximation for partially symmetric tensors (this approximation is also partially symmetric); rank- $$(r_1,\ldots ,r_d)$$ ( r 1 , ? , r d ) approximation for $$d$$ d -mode tensors.